On a Parabolic-Elliptic system with chemotaxis and logistic type growth

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• 作者：Evgeny Galakhov^a ; ^{galakhov@rambler.ru" class="auth_mail" title="E-mail the corresponding author} ; Olga Salieva^b ; J. Ignacio Tello^c ; ^d ; ^{j.tello@upm.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Parabolic&ndash ; Elliptic systems ; Global existence ; Chemotaxis ; Asymptotic behavior ; Stability
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：261
• 期：8
• 页码：4631-4647
• 全文大小：286 K

文摘

We consider a nonlinear PDEs system of two equations of Parabolic–Elliptic type with chemotactic terms. The system models the movement of a biological population “u” towards a higher concentration of a chemical agent “w  ” in a bounded and regular domain Ω⊂R^N$\mathrm{\Omega }\subset {\mathbb{R}}^N$ for arbitrary N∈N$N\in \mathbb{N}$. After normalization, the system is as follows for some positive constants m, χ, μ, α and γ  , with positive initial datum u₀$u_0$ and Neumann boundary conditions.

We study the range of parameters and constrains for which the solution exists globally in time. If either

α>m+γ−1$\alpha >m+\gamma -1$, or

α=m+γ−1$\alpha =m+\gamma -1$ and $\mu >\frac{N\alpha -2}{2(m-1)+N\alpha }\chi$,

the solution exists globally in time. Moreover, if and there exist positive constants ${\bar{u}}_0$ and 03f713f83622d150154e315">${\underset{\_}{u}}_0$ such that $0<{\underset{\_}{u}}_0\le u_0\le {\bar{u}}_0<\infty$ we have that